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Recently parabolic exhaustions have been used successfully to classify complex spaces (Stoll [6 and 7], Burns [2], P. Wong [9]). Here we define parabolic exhaustions for strictly convex domains in C and give necessary and sufficient conditions for such a domain to be biholomorphically equivalent to a circular domain or even to the ball. The results presented in this note are part of the Ph.D. Thesis the author is completing under the direction of Professor W. Stoll. Details and more implications will appear at a later date. A strictly convex domain D C C is a domain for which there exists a defining function whose real Hessian is strictly positive on Tx(dD) for all x G dD. Let A be the open unit disk in C and let S be the unit sphere in C . Let D c C be a strictly convex domain and p E D be any point. For b G S Lempert [4] constructs an extremal map F(D, b) : A —• D which is holomorphic with F(0,6) = v and F'(0,6) = X where 1/X > 0 is the length of b in the infinitesimal Kobayashi metric of D at p. These conditions determine F(D, b) uniquely and the map extends smoothly to an embedding F(U, b) : S —• D. Also F: A x S —• D is of class C°° and surjective. One and only one function r : D -> R+ exists such that r(F(z,b)) = \z\ for (z,b) G A X S. Then r is a continuous exhaustion of D, positive and of class C°° on D \ {p}. Also r = 1 on 3D. We refer to r as the Lempert exhaustion of D at p.

Characterization of strictly convex domains biholomorphic to a circular domain